Two moving coordinate frames for sweeping along a 3D trajectory

Abstract

In geometric modelling, translational and rotational sweeping are well known methods for defining the shape of 3D objects. The shape domain of sweeping can be extended by allowing objects to sweep along 3D trajectories that are represented by parametric polynomials (cubic splines, for instance), but which are otherwise arbitrary. The central problem is to define a triple of coordinates axes at each point of the trajectory in order to specify the position and orientation of the swept object along the trajectory. The resulting shape must be independent both of the parametrization and of the orientation of the trajectory in space. Two methods meeting these criteria are presented here. Geometric as well as computational properties of both methods are discussed, specifically for the case where a 2D closed curve is swept along the trajectory.